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Geometry of the Phase Retrieval Problem

Geometry of the Phase Retrieval Problem

Geometry of the Phase Retrieval Problem

Graveyard of Algorithms
Alexander H. Barnett, Flatiron Institute
Charles L. Epstein, Flatiron Institute
Leslie Greengard, Courant Institute
Jeremy Magland, Flatiron Institute
May 2022
Available
Hardback
9781316518878
NZD$198.95
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Hardback
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eBook

    Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications from astronomy to nanoscale X-ray diffraction imaging. Despite the efforts of a multitude of scientists, from astronomers to mathematicians, there is, as yet, no satisfactory theoretical or algorithmic solution to this class of problems. Written for mathematicians, physicists and engineers working in image analysis and reconstruction, this book introduces a conceptual, geometric framework for the analysis of these problems, leading to a deeper understanding of the essential, algorithmically independent, difficulty of their solutions. Using this framework, the book studies standard algorithms and a range of theoretical issues in phase retrieval and provides several new algorithms and approaches to this problem with the potential to improve the reconstructed images. The book is lavishly illustrated with the results of numerous numerical experiments that motivate the theoretical development and place it in the context of practical applications.

    • Features a careful analysis of the class of maps used in most algorithms, called hybrid iterative maps, including a complete description of the geometry underlying this class of maps that reveals many surprising properties
    • Equips the reader with tools to easily see when an algorithm provides reliable phase information for particular frequencies
    • Contains extensive background material on the mathematics employed in the book, making it accessible to a wide range of technically sophisticated physicists and engineers with an interest in phase retrieval
    • Includes nearly 200 color illustrations, including numerical examples whose results are displayed graphically

    Product details

    April 2022
    Adobe eBook Reader
    9781009008556
    0 pages
    This ISBN is for an eBook version which is distributed on our behalf by a third party.

    Table of Contents

    • Part I. Theoretical Foundations:
    • 1. The geometry near an intersection
    • 2. Well posedness
    • 3. Uniqueness and the non-negativity constraint
    • 4. Some preliminary conclusions
    • Part II. Analysis of Algorithms for Phase Retrieval:
    • 6. Introduction to Part II
    • 7. Algorithms for Phase Retrieval
    • 8. Discrete classical phase retrieval
    • 9. The non-negativity constraint
    • 10. Asymptotics of hybrid iterative maps
    • Part III. Further Properties of Hybrid Iterative Algorithms and Suggestions for Improvement:
    • 11. Introduction to Part III
    • 12. Statistics of algorithms
    • 13. Suggestions for improvements
    • 14. Concluding Remarks
    • 15. Notational conventions.
      Authors
    • Alexander H. Barnett , Flatiron Institute

      Alexander H. Barnett is Group Leader for Numerical Analysis at the Center for Computational Mathematics in the Flatiron Institute. He has published around 60 papers on partial differential equations, waves, fast algorithms, integral equations, neuroscience, imaging, signal processing, inverse problems, and physics, and received several research grants from the National Science Foundation.

    • Charles L. Epstein , Flatiron Institute

      Charles L. Epstein is the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, where he founded the graduate group in Applied Mathematics and Computational Science. He has worked on a wide range of problems in pure and applied analysis and is the author of a widely used textbook An Introduction to the Mathematics of Medical Imaging (SIAM 2008). He shared the Bergman Prize in 2016 with Francois Treves and is a fellow of the AAAS and the AMS.

    • Leslie Greengard , Courant Institute

      Leslie Greengard is Silver Professor of Mathematics and Computer Science at the Courant Institute, New York University and Director of the Center for Computational Mathematics at the Flatiron Institute. His is co-inventor of several widely used fast algorithms and a member of the American Academy of Arts and Sciences, the National Academy of Sciences, and the National Academy of Engineering.

    • Jeremy Magland , Flatiron Institute

      Jeremy Magland is a Senior Data Scientist at the Flatiron Institute. He received his PhD in Mathematics from the University of Pennsylvania. Prior to joining the Flatiron Institute in 2015, he worked for about a decade as a research scientist in the Radiology Department of the Hospital of the University of Pennsylvania, where he developed software systems that dramatically streamlined experimental work on MR-scanners.